Multi-database select overhead

Databases
1

Would a select that spans 2 (or even 3) databases consume more resources than a single-database query?
eg

SELECT db1.tab1.field1, db2.tab1.field1
FROM 
db1.tab1
JOIN db2.tab2 ON db1.tab1.field1 = db2.tab1.field2
2

Cross db joins are inherently heavy. You can benchmark it to know for sure, but it’s probably faster to query one and then run a secondary query off the initial result set.

3

Are both databases:

a) Located on the same physical server box?
b) Running from the same instance of MySQL?

4

Yes, and Yes